loos/eDFT_FUEG
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Manu: first polishing

Browse Source
This commit is contained in:
Emmanuel Fromager 2020-02-13 14:48:47 +01:00
parent cf45010298
commit 35d651e7e6
1 changed files with 141 additions and 594 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

733
Manuscript/eDFT.tex
View File

@ -92,7 +92,7 @@
%%% added by Manu %%% %%% added by Manu %%%
\newcommand{\manu}[1]{{\textcolor{blue}{ Manu: #1 }} } \newcommand{\manu}[1]{{\textcolor{darkgreen}{ Manu: #1 }} }
\newcommand{\beq}{\begin{eqnarray}} \newcommand{\beq}{\begin{eqnarray}}
\newcommand{\eeq}{\end{eqnarray}} \newcommand{\eeq}{\end{eqnarray}}
\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector \newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
@ -173,7 +173,7 @@ Atomic units are used throughout.
\label{sec:eDFT} \label{sec:eDFT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{GOK-DFT} \subsection{GOK-DFT}\label{subsec:gokdft}
The GOK ensemble energy is defined as follows: The GOK ensemble energy is defined as follows:
\beq \beq
@ -417,7 +417,7 @@ w}_K
%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%
\subsection{Approximations} \subsection{Approximations}\label{subsec:approx}
As Hartree and exchange energies cannot be separated in the As Hartree and exchange energies cannot be separated in the
one-dimension systems considered in the rest of this work, we will substitute the Hartree--Fock one-dimension systems considered in the rest of this work, we will substitute the Hartree--Fock
@ -493,472 +493,158 @@ c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}
\dfrac{\partial {\epsilon}^{{\bw}}_{\rm \dfrac{\partial {\epsilon}^{{\bw}}_{\rm
c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}. c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
\eeq \eeq
%%%% REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
%\iffalse%%%%
\blue{
Indeed,
\beq
\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
\nonumber\\
&=&\sum_{K\geq
0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
\nonumber\\
&=&
{\bmg}^{{\bw}}+\sum_{K,L\geq
0}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
\nonumber\\
&=&{\bmg}^{{\bw}}+w_0{\bmg}^{(0)}\times\sum_{K>0}w_K\left(2{\bmg}^{(K)}-1\right)
\nonumber\\
&&+\sum_{K, L >0
}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
\nonumber\\
&\neq&{\bmg}^{{\bw}}
.
\eeq
}
%%%% End -- REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
%\fi%%%
\blue{$================================$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory (old)} \section{Density-functional approximations for ensembles}
\label{sec:eDFT_old} \label{sec:eDFA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We decompose the weight-dependent functional as
\begin{equation}
\be{Hxc}{\bw}(\n{}{}) = \be{Hx}{\bw}(\n{}{}) + \be{c}{\bw}(\n{}{}),
\end{equation}
where $\be{Hx}{\bw}(\n{}{})$ is a weight-dependent Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} [see Subsec.~\ref{sec:GIC}] and $\be{c}{\bw}(\n{}{})$ is a weight-dependent correlation functional [see Subsec.~\ref{sec:Ec}].
The construction of these two functionals is described below.
Note that, because we consider strict 1D systems, one cannot decompose further the Hartree-exchange contribution as each component diverges independently but their sum is finite. \cite{Astrakharchik_2011, Lee_2011a, Loos_2012, Loos_2013, Loos_2013a}
Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (UEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
Moreover, because the infinite UEG model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
From this point of view, using finite UEGs \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems.
As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e., a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform.
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
This is a necessary condition for being able to model derivative discontinuities.
The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
\subsection{Kohn--Sham formulation of GOK-DFT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hybrid GOK-DFT} \subsection{Ghost-interaction correction}
\label{sec:geKS} \label{sec:GIC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\manu{I guess this subsection should be removed(?)}
Since Hartree and exchange energy contributions cannot be separated in \color{purple}
the one-dimensional case, we introduce in the following an alternative The GIC weight-dependent Hartree-exchange functional is defined as
formulation of KS-eDFT where, in complete analogy with the generalized \begin{multline}
KS scheme, a HF-like Hartree-exchange energy is employed. This \be{Hx}{\bw}(\n{}{\bw}) = (1-\sum_{I>0} \ew{I}) \be{Hx}{}(\n{}{(0)}) + \sum_{I>0} \ew{I} \be{Hx}{}(\n{}{(I)})
formulation is in principle exact and applicable to higher dimensions. \\
- \be{Hx}{(I)}(\n{}{\bw}),
When $\bw=0$, the \end{multline}
conventional ground-state universal functional is recovered, where
\beq \begin{equation}
F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min} \be{Hx}{}(\n{}{}) = \iint \frac{\n{}{}(\br_1) \n{}{}(\br_2) - \n{}{}(\br_1,\br_2)^2}{r_{12}} d\br_1 d\br_2,
\bra{\Psi}\hat{T}+\hat{W}_{\rm \end{equation}
ee}\ket{\Psi},
\eeq
where the ensemble reduces to a single wavefunction. In the latter case,
the HF-like expression (or a fraction of it, as usually done in
practical calculations) for the Hx energy can be introduced rigorously
into DFT by considering the following decomposition,
\beq\label{eq:generalized_KS-DFT_decomp}
F[n]&=&
\underset{\Phi\rightarrow n}{\rm min}
\bra{\Phi}\hat{T}+\hat{W}_{\rm
ee}\ket{\Phi}+\overline{E}_{\rm c}[n]
\nonumber\\
&=&
\underset{\bmg^\Phi\rightarrow n}{\rm min}
\left\{{\rm
Tr}\left[\bmg^\Phi{\bm t}\right]+W_{\rm
HF}\left[{\bmg}^{\Phi}\right]\right\}+
\overline{E}_{\rm c}[n]
,
\eeq
where ${\bm t}$ is the matrix representation of the one-electron kinetic
energy operator, $\bmg^\Phi$ is the one-electron reduced density
matrix (just referred to as density matrix in the following) obtained
from $\Phi$,
and and
\beq \begin{equation}
W_{\rm \n{}{(I)}(\omega) = (\pi R)^{-1} \cos[(I+1) \omega/2]
HF}\left[{\bmg}\right]\equiv\frac{1}{2} \Tr(\bmg \, \bG \, \bmg) \end{equation}
\eeq is the first-order density matrix with $\omega$ the interelectronic angle.
is the conventional density-matrix functional HF Hartree-exchange It yields
energy. By analogy with Eq.~(\ref{eq:generalized_KS-DFT_decomp}), we \begin{equation}
decompose the ensemble universal functional as follows: \be{Hx}{}(\n{}{}) = \n{}{} \qty[ a_1 \ew{1} (\ew{1} - 1) + a_2 \ew{1} \ew{2} + a_3 \ew{2} (\ew{2} - 1)],
\beq\label{eq:generalized_F_w} \end{equation}
F^{\bw}[n]&=& with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
\underset{\hat{\Gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm \color{black}
Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+W_{\rm \subsection{Weight-dependent correlation functional}
HF}\left[{\bmg}^{\bw}\right]\right\} \label{sec:Ec}
+\overline{E}^{{\bw}}_{\rm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Hxc}[n]
\nonumber\\
&=&
\underset{{\bmg}^{{\bw}}\rightarrow n}{\rm min}
\left\{
{\rm Tr}
\left[{\bmg}^{\bw}{\bm t}\right]
+W_{\rm HF}\left[{\bmg}^{\bw}\right]
\right\}+
\overline{E}^{\bw}_{\rm Hxc}[n],
\eeq
where the minimization in Eq.~(\ref{eq:ens_LL_func}) has been restricted
to density matrix operators
\beq
\hat{\Gamma}^{{\bw}}=\sum_{K\geq 0}w_K\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum_{K\geq 0}w_K\hat{\Gamma}^{(K)}
\eeq
that are constructed from single Slater
determinants $\Phi^{(K)}$. Note that the density matrices
${\bmg}^{(K)}={\bmg}^{\Phi^{(K)}}$ are idempotent and diagonal in the
same spin-orbital basis). On the other hand, the ensemble
density matrix ${\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}$, which is a convex combination of the ${\bmg}^{(K)}$
matrices, is {\it not} idempotent, unless ${\bw}=0$.
Using an ensemble is, in this context,
analogous to assigning
fractional occupation numbers (which are determined from the ensemble
weights) to the KS orbitals.\\
Another issue with the use of Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
ensembles in DFT is the introduction of spurious ghost-interaction errors \begin{equation}
(i.e. unphysical interactions between different states) into the \label{eq:ec}
ensemble energy when inserting ${\bmg}^{{\bw}}$ into the HF \e{c}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
density-matrix functional Hx energy $W_{\rm \end{equation}
HF}\left[\bmg\right]$. This type of errors is specific to ensembles where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
which explains why, in constrast to ground-state DFT [see The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
Eq.~(\ref{eq:generalized_KS-DFT_decomp})], a complementary ensemble Hx Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
energy is needed to recover a ghost-interaction-free energy: Combining these, one can build a three-state weight-dependent correlation eDFA:
\beq\label{eq:exact_GIC} \begin{equation}
\overline{E}^{{\bw}}_{\rm \label{eq:ecw}
Hx}[n]&=& \e{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
{\rm \end{equation}
Tr}\left[\hat{\Gamma}^{{\bw}}[n]\hat{W}_{\rm ee}\right]-W_{\rm
HF}\left[{\bmg}^{\bw}[n]\right]
\nonumber\\
&=&
\sum_{K\geq0}w_KW_{\rm
HF}\left[{\bmg}^{(K)}[n]\right]
-W_{\rm
HF}\left[{\bmg}^{\bw}[n]\right],
\eeq
where ${\bmg}^{\bw}[n]$ is the minimizing ensemble density matrix in
Eq.~(\ref{eq:generalized_F_w}) and, by construction, $\overline{E}^{{\bw}=0}_{\rm
Hx}[n]=0$. Consequently, the ensemble correlation functional can be
expressed as follows [see Eq.~(\ref{eq:generalized_F_w})]:
\beq
\overline{E}^{{\bw}}_{\rm
c}[n]&=&
\overline{E}^{\bw}_{\rm Hxc}[n]-\overline{E}^{{\bw}}_{\rm
Hx}[n]
\nonumber\\
&=&{\rm
Tr}\left[\hat{\gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
ee}\right)\right]
%\nonumber\\
%&&
-
{\rm
Tr}\left[\hat{\Gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
ee}\right)\right]
\nonumber\\
&=&
\sum_{K\geq 0}w_K\Bigg(\bra{\Psi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
ee}\ket{\Psi^{(K)}[n]}
\nonumber\\
&&-\bra{\Phi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
ee}\ket{\Phi^{(K)}[n]}\Bigg),
\eeq
where $\hat{\gamma}^{{\bw}}[n]$ and $\hat{\Gamma}^{{\bw}}[n]$ are the minimizing density matrix
operators in Eqs.~(\ref{eq:ens_LL_func}) and
(\ref{eq:generalized_KS-DFT_decomp}), respectively.\\
%%% TABLE 1 %%%
\begin{table*}
\caption{
\label{tab:OG_func}
Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
% \begin{ruledtabular}
\begin{tabular}{lcddd}
\hline\hline
State & $I$ & \tabc{$c_1^{(I)}$} & \tabc{$c_2^{(I)}$} & \tabc{$c_3^{(I)}$} \\
\hline
Ground state & $0$ & -0.0137078 & 0.0538982 & 0.0751740 \\
Singly-excited state & $1$ & -0.0238184 & 0.00413142 & 0.0568648 \\
Doubly-excited state & $2$ & -0.00935749 & -0.0261936 & 0.0336645 \\
\hline\hline
\end{tabular}
% \end{ruledtabular}
\end{table*}
%%% %%% %%% %%%
In eDFT, the ensemble energy $E^{{\bw}}=\sum_{K\geq %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0}w_KE^{(K)}$ is obtained variationally as follows: \subsection{LDA-centered functional}
\beq %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E^{{\bw}}=\underset{n}{\rm min}\Big\{ In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
F^{\bw}[n]+\int d\br\,v_{\rm ext}(\br)n(\br) \begin{equation}
\Big\}. \label{eq:becw}
\eeq \be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
Combining the latter expression with the decomposition in \end{equation}
Eq.~(\ref{eq:generalized_KS-DFT_decomp}) leads to where
\beq \begin{equation}
E^{{\bw}}= \be{c}{(I)}(\n{}{}) = \e{c}{(I)}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}) - \e{c}{(0)}(\n{}{}).
\underset{n}{\rm min}\Bigg\{ \end{equation}
\underset{{\bmg}^{{\bw}}\rightarrow n}{\rm min}\Big\{ The local-density approximation (LDA) correlation functional,
{\rm \begin{equation}
Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm \e{c}{\text{LDA}}(\n{}{}) = c_1^\text{LDA} \, F\qty[1,\frac{3}{2},c_3^\text{LDA}, \frac{c_1^\text{LDA}(1-c_3^\text{LDA})}{c_2^\text{LDA}} {\n{}{}}^{-1}],
HF}\left[{\bmg}^{\bw}\right] \end{equation}
+ specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
\overline{E}^{{\bw}}_{\rm \begin{align}
Hxc}\left[n_{\bmg^{\bw}}\right] c_1^\text{LDA} & = - \frac{\pi^2}{360},
%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right] &
\Big\} c_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
\Bigg\} &
\nonumber\\ c_3^\text{LDA} & = 2.408779.
\eeq \end{align}
or, equivalently, Equation \eqref{eq:becw} can be recast
\beq\label{eq:var_princ_Gamma_ens} \begin{equation}
E^{{\bw}}= \label{eq:eLDA}
\underset{{\bmg}^{{\bw}}}{\rm min}\Big\{ \begin{split}
{\rm \be{c}{\bw}(\n{}{})
Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm & = \e{c}{\text{LDA}}(\n{}{})
HF}\left[{\bmg}^{\bw}\right] \\
+ & + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
\overline{E}^{{\bw}}_{\rm \end{split}
Hxc}\left[n_{\bmg^{\bw}}\right] \end{equation}
%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right] which nicely highlights the centrality of the LDA in the present eDFA.
\Big\} In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
, Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
\eeq
where $n_{\bmg^{\bw}}$ is the density obtained from the density matrix
${\bmg}^{\bw}$ and ${\bm h}={\bm t}+{\bm v}_{\rm ext}$ is the total one-electron
Hamiltonian matrix representation. When the minimum is reached, the
ensemble energy and its derivatives can be used to extract individual
ground- and excited-state energies as follows:\cite{Deur_2018b}
\beq
E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w_K\right)\dfrac{\partial
E^{{\bw}}}{\partial w_K}.
\eeq
Since, according to the Hellmann--Feynman theorem, the ensemble energy
derivative reads
\beq
\dfrac{\partial E^{{\bw}}}{\partial w_K}&=&{\rm
Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
\nonumber\\
&&+\Tr\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right) \, \bG \, \bmg^{\bw}\right]
\nonumber\\
&&+
\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\left(n_{\bmg^{(K)}}(\br)-n_{\bmg^{(0)}}(\br)\right)
\nonumber\\
&&
+\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
Hxc}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
,
\eeq
we finally obtain from Eqs.~(\ref{eq:var_princ_Gamma_ens}) and (\ref{eq:indiv_ener_from_ens}) the following in-principle-exact expressions for the
energy levels within the ensemble:
\beq
&&E^{(I)}=
{\rm
Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
\Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
\nonumber\\
&&+\overline{E}^{{\bw}}_{\rm
Hxc}\left[n_{\bmg^{\bw}}\right]
+\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
\nonumber\\
&&
+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
Hxc}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}.
\eeq
%+\Tr(\bmg^{(I)} \, \bG \, \bmg^{\bw})
%-\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})+...
\alert{
Note that
\beq
\overline{E}^{{\bw}}_{\rm
Hx}\left[n_{\bmg^{\bw}}\right]=
\frac{1}{2} \sum_{L\geq0}w_L \Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
-\frac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})
\nonumber\\
\eeq
and
\beq
\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
Hx}[n]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}&=&
\frac{1}{2} \Tr(\bmg^{(K)} \, \bG \, \bmg^{(K)})-\frac{1}{2}
\Tr(\bmg^{(0)} \, \bG \, \bmg^{(0)})
\nonumber\\
&&-\Tr\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right) \, \bG \, \bmg^{\bw}\right]
+\ldots
\eeq
thus leading to
\beq
&&\overline{E}^{{\bw}}_{\rm
Hx}\left[n_{\bmg^{\bw}}\right]+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
Hx}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
\nonumber\\
&&=
\overline{E}^{{\bw}}_{\rm
Hx}\left[n_{\bmg^{\bw}}\right]+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \, \bmg^{(I)})
-\frac{1}{2} \sum_{L\geq0}w_L \Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
\nonumber\\
&&-\Tr\left[\left({\bmg}^{(I)}-{\bmg}^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
+\ldots
\nonumber\\
&&=\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
\bmg^{(I)})-\Tr\left[\left({\bmg}^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right)
\, \bG \, \bmg^{\bw}\right]
+\ldots
\eeq
}
At the eLDA level:
\beq This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014}
\overline{E}^{{\bw}}_{\rm Within this in-principle-exact formalism, the (weight-dependent) correlation energy of the ensemble is constructed from the (weight-independent) ground-state functional (such as the LDA), yielding Eq.~\eqref{eq:eLDA}.
Hxc}\left[n\right]\rightarrow\int d\br\,n(\br)\overline{\epsilon}^{{\bw}}_{\rm This is a crucial point as we intend to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons).
Hxc}(n(\br))
\eeq
\beq
\dfrac{\delta \overline{E}^{{\bw}}_{\rm
Hxc}\left[n\right]}{\delta
n({\br})}\rightarrow \overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n(\br))+n(\br)\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n)}{\partial n}\right|_{n=n(\br)}
\eeq
\beq Finally, we note that, by construction,
&&E^{(I)}\rightarrow \begin{equation}
{\rm \left. \pdv{\be{c}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br)} = \be{c}{(J)}[\n{}{\bw}(\br)] - \be{c}{(0)}[\n{}{\bw}(\br)].
Tr}\left[{\bmg}^{(I)}{\bm h}\right]+ \end{equation}
\Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right] \alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
\nonumber\\
&&
+\int d\br\,
\overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
\nonumber\\
&&
+\int d\br\,\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
\nonumber\\
&&
+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
\eeq
\alert{
or, equivalently,
\beq
&&E^{(I)}\rightarrow
{\rm
Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
\Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
\nonumber\\
&&
+\int d\br\,
\dfrac{\delta \overline{E}^{{\bw}}_{\rm
Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\,n_{\bmg^{(I)}}(\br)
\nonumber\\
&&
-\int d\br\,\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}\Big(n_{\bmg^{\bw}}(\br)\Big)^2
\nonumber\\
&&
+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
\eeq
}
\subsection{Exact ensemble exchange in hybrid GOK-DFT}
In the exact theory, the minimizing density matrix in
Eq.~(\ref{eq:var_princ_Gamma_ens}) is such that
\beq
{\bmg}^{(K)}[n_{{\bmg}^{{\bw}}}]={\bmg}^{(K)},\hspace{0.2cm}\forall
K\geq0,
\eeq
and therefore
\beq
{\bmg}^{{\bw}}\left[n_{{\bmg}^{{\bw}}}\right]={\bmg}^{{\bw}}.
\eeq
Combining the latter Eqs. with
Eqs. (\ref{eq:exact_GIC}), (\ref{eq:var_princ_Gamma_ens}) leads to
the final ensemble energy expression
\beq\label{eq:exact_Eens_EEXX}
E^{{\bw}}={\rm
Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+\frac{1}{2} \sum_{L\geq0}w_L
\Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
+\overline{E}^{{\bw}}_{\rm
c}\left[n_{\bmg^{\bw}}\right].
\nonumber\\
\eeq
Note that
\beq
E^{{\bw}}&\neq& \underset{\left\{{\bmg}^{(L)}\right\}_{L\geq 0}}{\rm min}\Big\{
{\rm
Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+\frac{1}{2} \sum_{L\geq0}w_L
\Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
\nonumber\\
&&+\overline{E}^{{\bw}}_{\rm
c}\left[n_{\bmg^{\bw}}\right]
\Bigg\}
\eeq
For $K>0$
\beq\label{eq:XE_EEXX}
&&\dfrac{\partial E^{{\bw}}}{\partial w_K}=
{\rm
Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
\nonumber\\
&&+\frac{1}{2}\Tr(\bmg^{(K)} \, \bG \, \bmg^{(K)})
-\frac{1}{2}\Tr(\bmg^{(0)} \, \bG \, \bmg^{(0)})
\nonumber\\
&&
+\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
c}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\left(n_{\bmg^{(K)}}(\br)-n_{\bmg^{(0)}}(\br)\right)
+\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
\nonumber\\
&&+\sum_{L\geq0}w_L{\rm
Tr}\left[\dfrac{\partial\bmg^{(L)}}{\partial w_K}{\bm h}\right]
%\nonumber\\
%&&
+\sum_{L\geq0}w_L
\Tr(\bmg^{(L)} \, \bG \, \dfrac{\partial\bmg^{(L)}}{\partial w_K})
\nonumber\\
&&
+\sum_{L\geq0}w_L\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
c}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}n_{\frac{\partial \bmg^{(L)}}{\partial w_K}}(\br).
\eeq
If we introduce individual Fock matrices
\beq
{\bm F}^{(L)}={\bm h}+\bG \,\bmg^{(L)}+\overline{\bm v}^{{\bw}}_{\rm
c}\left[n_{\bmg^{\bw}}\right],
\eeq
the last three terms can be simply rewritten as
\beq
\sum_{L\geq0}w_L{\rm
Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right].
\eeq
According to Eqs.~(\ref{eq:indiv_ener_from_ens}),
(\ref{eq:exact_Eens_EEXX}), and (\ref{eq:XE_EEXX}),
\beq\label{eq:exact_ind_ener_OEP-like}
E^{(I)}&&={\rm
Tr}\left[{\bmg}^{(I)}{\bm h}\right]
+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
\bmg^{(I)})
\nonumber\\
&&+\overline{E}^{{\bw}}_{\rm
c}\left[n_{\bmg^{\bw}}\right]
+\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
c}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
\nonumber\\
&&+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
\nonumber\\
&&
+\sum_{K>0}\left(\delta_{IK}-w_K\right)\sum_{L\geq0}w_L{\rm
Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right]
.
\eeq
\subsection{Ensemble
correlation LDA and ghost interaction correction
}
\alert{Secs. \ref{sec:KS-eDFT}-\ref{sec:E_I} should maybe be moved to an appendix or merged
with the theory section (?)}
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
\color{purple}
\section{Implementation} \section{Implementation}
\manu{I think that this section can be removed (especially Sec.~\ref{sec:E_I}). Many points
discussed in Sec.~\ref{sec:KS-eDFT} are now mentioned in the theory
section. If we want to keep some material of Sec.~\ref{sec:KS-eDFT}, it
should be moved to
Secs.~\ref{subsec:gokdft} or~\ref{subsec:approx} (or maybe Sec.~\ref{sec:compdetails}).}
%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -1051,148 +737,9 @@ The (state-independent) Levy-Zahariev shift and the so-called derivative discont
\end{align} \end{align}
Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}]. Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}].
The only remaining piece of information to define at this stage is the weight-dependent Hartree-exchange-correlation functional $\be{Hxc}{\bw}(\n{}{})$. The only remaining piece of information to define at this stage is the weight-dependent Hartree-exchange-correlation functional $\be{Hxc}{\bw}(\n{}{})$.
\color{black}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Density-functional approximations for ensembles} \section{Computational details}\label{sec:compdetails}
\label{sec:eDFA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We decompose the weight-dependent functional as
\begin{equation}
\be{Hxc}{\bw}(\n{}{}) = \be{Hx}{\bw}(\n{}{}) + \be{c}{\bw}(\n{}{}),
\end{equation}
where $\be{Hx}{\bw}(\n{}{})$ is a weight-dependent Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} [see Subsec.~\ref{sec:GIC}] and $\be{c}{\bw}(\n{}{})$ is a weight-dependent correlation functional [see Subsec.~\ref{sec:Ec}].
The construction of these two functionals is described below.
Note that, because we consider strict 1D systems, one cannot decompose further the Hartree-exchange contribution as each component diverges independently but their sum is finite. \cite{Astrakharchik_2011, Lee_2011a, Loos_2012, Loos_2013, Loos_2013a}
Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (UEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
Moreover, because the infinite UEG model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
From this point of view, using finite UEGs \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems.
As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e., a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform.
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
This is a necessary condition for being able to model derivative discontinuities.
The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ghost-interaction correction}
\label{sec:GIC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The GIC weight-dependent Hartree-exchange functional is defined as
\begin{multline}
\be{Hx}{\bw}(\n{}{\bw}) = (1-\sum_{I>0} \ew{I}) \be{Hx}{}(\n{}{(0)}) + \sum_{I>0} \ew{I} \be{Hx}{}(\n{}{(I)})
\\
- \be{Hx}{(I)}(\n{}{\bw}),
\end{multline}
where
\begin{equation}
\be{Hx}{}(\n{}{}) = \iint \frac{\n{}{}(\br_1) \n{}{}(\br_2) - \n{}{}(\br_1,\br_2)^2}{r_{12}} d\br_1 d\br_2,
\end{equation}
and
\begin{equation}
\n{}{(I)}(\omega) = (\pi R)^{-1} \cos[(I+1) \omega/2]
\end{equation}
is the first-order density matrix with $\omega$ the interelectronic angle.
It yields
\begin{equation}
\be{Hx}{}(\n{}{}) = \n{}{} \qty[ a_1 \ew{1} (\ew{1} - 1) + a_2 \ew{1} \ew{2} + a_3 \ew{2} (\ew{2} - 1)],
\end{equation}
with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent correlation functional}
\label{sec:Ec}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
\begin{equation}
\label{eq:ec}
\e{c}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
\end{equation}
where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
Combining these, one can build a three-state weight-dependent correlation eDFA:
\begin{equation}
\label{eq:ecw}
\e{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
\end{equation}
%%% TABLE 1 %%%
\begin{table*}
\caption{
\label{tab:OG_func}
Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
% \begin{ruledtabular}
\begin{tabular}{lcddd}
\hline\hline
State & $I$ & \tabc{$c_1^{(I)}$} & \tabc{$c_2^{(I)}$} & \tabc{$c_3^{(I)}$} \\
\hline
Ground state & $0$ & -0.0137078 & 0.0538982 & 0.0751740 \\
Singly-excited state & $1$ & -0.0238184 & 0.00413142 & 0.0568648 \\
Doubly-excited state & $2$ & -0.00935749 & -0.0261936 & 0.0336645 \\
\hline\hline
\end{tabular}
% \end{ruledtabular}
\end{table*}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{LDA-centered functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
\begin{equation}
\label{eq:becw}
\be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
\end{equation}
where
\begin{equation}
\be{c}{(I)}(\n{}{}) = \e{c}{(I)}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}) - \e{c}{(0)}(\n{}{}).
\end{equation}
The local-density approximation (LDA) correlation functional,
\begin{equation}
\e{c}{\text{LDA}}(\n{}{}) = c_1^\text{LDA} \, F\qty[1,\frac{3}{2},c_3^\text{LDA}, \frac{c_1^\text{LDA}(1-c_3^\text{LDA})}{c_2^\text{LDA}} {\n{}{}}^{-1}],
\end{equation}
specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
\begin{align}
c_1^\text{LDA} & = - \frac{\pi^2}{360},
&
c_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
&
c_3^\text{LDA} & = 2.408779.
\end{align}
Equation \eqref{eq:becw} can be recast
\begin{equation}
\label{eq:eLDA}
\begin{split}
\be{c}{\bw}(\n{}{})
& = \e{c}{\text{LDA}}(\n{}{})
\\
& + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
\end{split}
\end{equation}
which nicely highlights the centrality of the LDA in the present eDFA.
In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014}
Within this in-principle-exact formalism, the (weight-dependent) correlation energy of the ensemble is constructed from the (weight-independent) ground-state functional (such as the LDA), yielding Eq.~\eqref{eq:eLDA}.
This is a crucial point as we intend to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons).
Finally, we note that, by construction,
\begin{equation}
\left. \pdv{\be{c}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br)} = \be{c}{(J)}[\n{}{\bw}(\br)] - \be{c}{(0)}[\n{}{\bw}(\br)].
\end{equation}
\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation. Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\Nel$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\Nel$-boxium. Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\Nel$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\Nel$-boxium.